I personally would never say "I shouldn't like to be in his shoes" - "I should not like to be in his shoes" - why should you not? You either want to be in his shoes or you don't. "I wouldn't like to be in his shoes" - "I would not like..........." means I'd not like to be in that situation. "I would like to be in his shoes" - if he won millions of pounds, of course you'd like to be in his shoes. So, why would a person say "I shouldn't like to be in his shoes" - there would have to be some form of negative comeback or punishment for liking the idea of being in his shoes. Does that make sense?? It would only make sense to me if say for example it represented a type of unpleasant consequence - "his poor wife has just died, he has to leave the company, that means his job is free now, I am happy I have his job - but when I think about the circumstances, I shouldn't like to be in his shoes. This is sad for him" But I'd never swap wouldn't for shouldn't in this example. Good question though, again!

They have the same meaning, but "I shouldn't like to be in his shoes" sounds more British. In modern American English we'd say "I wouldn't like to be in his shoes." So it just depends on how you want to sound.